If $X$ is a measure space compatible with an underlying topological structure, what is the dual space of $L^p _{loc} (X)$ (the space of locally $p$-integrable functions) for $p \in [1,\infty)$? When $p=1$, a good answer has already been provided, but I am looking for an argument emphasizing the structure of inductive limit space of $L^p _{loc} (X)$.
I was thinking that $L^p _{loc} (X) \simeq \varinjlim \limits _K L^p (K)$ (where $K$ runs over the compacts of $X$) and taking into consideration that, in general, $\hom(\varinjlim X_i , Y) \simeq \varprojlim \hom(X_i, Y)$, I could write
$$L^p _{loc} (X) ^* = \hom \big(\varinjlim \limits _K L^p (K), \Bbb C \big) \simeq \varprojlim \limits _K \hom \big( L^p (K), \Bbb C \big) = \varprojlim \limits _K L^p (K) ^* = \varprojlim \limits _K L^q (K) \ ,$$
where $\frac 1 q + \frac 1 p = 1$, but how do I continue from here (if possible to continue at all)?
In this proof I shall consider already known the fact that the topology on $L^p _{loc} (X)$ is given by the family of seminorms $$f \mapsto p_K (f) = \| f \big| _K \| _{L^p (K)}$$ for all compact subsets $K \subseteq X$ and all $f \in L^p _{loc} (X)$.
Let $\omega \in L^p _{loc} (X) ^*$. This means that there exist $c_1, \dots, c_m > 0$ and compact subsets $K_1, \dots, K_m$ such that $$|\omega (f)| \le c_1 \ p_{K_1} (f) + \dots + c_m \ p_{K_m} (f)$$ for all $f \in L^p _{loc} (X)$. If $c = \max \{c_1, \dots, c_m\}$ and $K = K_1 \cup \dots K_m$, the above implies that $$|\omega (f)| \le c \ p_K (f) \ .$$ Let us show that $\omega$ factorizes through $L^p (K)$. Indeed, if $r_K : L^p _{loc} (X) \to L^p (X)$ is the restriction $r_K (f) = f \big| _K$, then the above inequality implies that $\ker r_k \subseteq \ker \omega$, therefore there exists $\omega_K \in L^p (K) ^*$ such that $\omega = \omega_K \circ r_K$. But we know that $L^p (K)^* \simeq L^q (K)$ where $q = \frac p {p-1}$, which means that there exists $\varphi \in L^q (K)$ such that $\omega_K (g) = \int _K \varphi \ g \ \mathrm d \mu$ for all $g \in L^p (K)$, whence $$\omega (f) = \int _K \varphi \ f \big| _K \ \mathrm d \mu = \int _X \widetilde \varphi \ f \ \mathrm d \mu \ ,$$ where $\tilde f$ is the extension of $f$ by $0$ on $X \setminus K$.
We have shown that each functional in $L^p _{loc} (X) ^*$ is given by integration against some function from $L^q (X)$ having compact essential support.